Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-11T06:58:56.519Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Digits of Sumsets

Published online by Cambridge University Press:  20 November 2018

Christian Mauduit ,
Joël Rivat  and
András Sárközy
Christian Mauduit
Affiliation:
Université d'Aix-Marseille and Institut Universitaire de France, Institut de Mathématiques de Marseille, CNRS UMR 7373, 163, avenue de Luminy, Case 9, 13288 MARSEILLE Cedex 9, France e-mail: mauduit@iml.univ-mrs.fr
Joël Rivat
Affiliation:
Université d'Aix-Marseille, Institut de Mathématiques de Marseille, CNRS UMR 7373, 163, avenue de Luminy, Case 907, 13288 MARSEILLE Cedex 9, France e-mail: joel.rivat@univ-amu.fr
András Sárközy
Affiliation:
Eötvös Loránd University, Department of Algebra and Number Theory, H-1117 Budapest, Pázmány Péter sétány 1/c, Hungary e-mail: sarkozy@cs.elte.hu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\mathcal{A}$ and $\mathcal{B}$ be large subsets of $\{1,\,.\,.\,.\,,\,N\}$. We study the number of pairs $\left( a,b \right)\,\in \,\mathcal{A}\,\times \,\mathcal{B}$ such that the sum of binary digits of $a\,+\,b$ is fixed.

Keywords

11A6311B13sumsetdigits
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 3 , 01 June 2017 , pp. 595 - 612
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Alon, N. and Spencer, J. H., The probabilistic method. Third ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008.http://dx.doi.org/10.1002/9780470277331 Google Scholar
[2] Balog, A., Rivat, J., and Sárközy, A., On arithmetic properties ofsumsets. Acta Math. Hungar. 144(2014), no. 1, 18–42.http://dx.doi.Org/10.1007/s10474-014-0436-y Google Scholar
[3] Drmota, M., Subsequences of automatic sequences and uniform distribution. In: Uniform distribution and quasi-Monte Carlo methods, Radon Ser. Comput. Appl. Math., 15, De Gruyter, Berlin, 2014, pp. 87–104.Google Scholar
[4] Drmota, M., Mauduit, C., and Rivat, J., The sum-of-digits function of polynomial sequences. J. Lond. Math. Soc. (2) 84(2011), no. 1, 81–102.http://dx.doi.org/10.1112/jlms/jdrOO3 Google Scholar
[5] Drmota, M. and Morgenbesser, J. F., Generalized Thue-Morse sequences of squares. Israel J. Math. 190(2012), 157–193.http://dx.doi.org/10.1007/s11856-011-0186-2 Google Scholar
[6] Fouvry, E. and Mauduit, C., Sur les entiers dont la somme des chiffres est moyenne. J. Number Theory 114(2005), no. 1, 135–152. http://dx.doi.Org/10.1016/j.jnt.2005.03.007 Google Scholar
[7] Gel'fond, A. O., Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13(1967/1968), 259–265. Google Scholar
[8] Martin, B., Mauduit, C., and Rivat, J., Théoréme des nombres premiers pour les fonctions digitales. Acta Arith. 165(2014), no. 1, 11–45.http://dx.doi.org/10.4064/aa165-1-2 Google Scholar
[9] Martin, B., Fonctions digitales le long des nombres premiers. Acta Arith. 170(2015), no. 2,175–197. http://dx.doi.org/10.4064/aa170-2-5 Google Scholar
[10] Mauduit, C., Propriétés arithmétiques des substitutions et automates infinis. Ann. Inst. Fourier 56(2006), no. 7, 2525–2549.http://dx.doi.Org/10.5802/aif.2248 Google Scholar
[11] Mauduit, C. and Moreira, C. G., Phénoméne de Moser-Newman pour les nombres sans facteur carré. Bull. Soc. Math. France 143(2015), no. 3, 599–617.Google Scholar
[12] Mauduit, C., Pomerance, C., and Sárközy, A., On the distribution in residue classes of integers with a fixed sum of digits. Ramanujan J. 9(2005), no. 1-2, 45–62.http://dx.doi.Org/10.1007/s11139-005-0824-6 Google Scholar
[13] Mauduit, C. and Rivat, J., Propriétés q-multiplicatives de la suite [nc╛, c > 1. Acta Arith. 118(2005), no. 2, 187–203.http://dx.doi.org/10.4064/aa118-2-6 +1.+Acta+Arith.+118(2005),+no.+2,+187–203.http://dx.doi.org/10.4064/aa118-2-6>Google Scholar
[14] Mauduit, C., La somme des chiffres des carrés. Acta Math. 203(2009), no. 1,107–148.http://dx.doi.Org/10.1007/s11511-009-0040-0 Google Scholar
[15] Mauduit, C., Sur un probléme de Gelfond: la somme des chiffres des nombres premiers. Ann. of Math. (2) 171(2010), no. 3, 1591–1646.http://dx.doi.Org/10.4007/annals.2010.1 71.1591 Google Scholar
[16] Mauduit, C. and Sárközy, A., On the arithmetic structure of sets characterized by sum of digits properties. J. Number Theory 61(1996), no. 1, 25–38.http://dx.doi.Org/10.1006/jnth.1996.0134 Google Scholar
[17] Mauduit, C., On the arithmetic structure of the integers whose sum of digits is fixed. Acta Arith. 81(1997), no. 2, 145–173.Google Scholar
[18] Newman, D. J. and Slater, M., Binary digit distribution over naturally defined sequences. Trans. Amer. Math. Soc. 213(1975), 71–78.http://dx.doi.Org/10.1090/S0002-9947-1 975-0384734-3 Google Scholar
[19] Spiegelhofer, L., Piatetski-Shapiro sequences via Beatty sequences. Acta Arith. 166(2014), no. 3, 201–229.http://dx.doi.org/10.4064/aa166-3-1 Google Scholar